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Abstract

This paper deals with the Hausdorff dimension of the Julia set of quadratic polynomials. It is divided in two parts. The first aims to compute good numerical approximations of the dimension for hyperbolic points. For such points, Ruelle's thermodynamical formalism applies, hence computing the dimension amounts to computing the zero point of a pressure function. It is this pressure function that we approximate by a Monte-Carlo process combined with a shift method that considerably decreases the computational cost. The second part is a continuity result of the dimension on the real axis at the parabolic point 1/4 for $P_c(z)=z^2+c$.